How many pairs of co-prime natural numbers a and b exist such that a + b = 510

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Question

How many pairs of co-prime natural numbers a and b exist such that a + b = 510?

Options:

A . 60

B . 120

C . 64

D . 128

Answer: Option C
:
C

$510 = 2 \times 3 \times 5 \times 17$ .
Use the concept of Euler's theorem:
$510 (1 - \frac{1}{2}) (1 - \frac{1}{3}) (1 - \frac{1}{5}) (1 - \frac{1}{17}) = 128$
$N u m b e r o f p a i r s = \frac{128}{2} = 64$ .

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